Mathematics Question Review (2025-10-01)
Question:
Denote a sequence {a_n}
= 1 + 1/2 + 1/3 + ... + 1/n - ln(n),
where n = 1, 2, 3 ....
Show that {a_n} converges.
答案聽日開估
Topic: Limit of a sequence (HKAL)
Every NDS and PSP from graduate entrance exams have learnt this topic already.
Hints:
Note: x/(x+1) = 1- 1/(x+1)
Second hint:
Third hint:
ln(1+1) + ln(1+1/2) + ... + ln(1+1/n) = ln(n+1)
Now 0 < 1/(n+1) < ln(1+ 1/n) < 1/n
∑ 1/(n+1) < ∑ ln(1+ 1/n) = ln(n+1)
and -ln(n) < ln(1/n) < 0 <1/n
Then ln(n+1) + 1/n = ln(n+1) + (-1/n) > 0
Now we manage to show that {a_n} has lower bound 0 and is strictly decreasing as n increases, so the sequence converges. Q.E.D.
嗱，HKAL 純數望落好難，但落手落腳做嘅時候，
好多時都係講緊gimmicks同rules of thumb，
肯操就肯定合格，夠小心夠快計完數就 aim ABC
當然如果你成日犯基本manipulation mistakes，
就同烏大龜計數慢一樣，與大學無緣
Now 0 < 1/(n+1) < ln(1+ 1/n) < 1/n
∑ 1/(n+1) < ∑ ln(1+ 1/n) = ln(n+1)
and -ln(n) < ln(1/n) < 0 <1/n
Then ln(n+1) + 1/n = ln(n+1) + (-1/n) > 0
Now we manage to show that {a_n} has lower bound 0 and is strictly decreasing as n increases, so the sequence converges. Q.E.D.
ln(n+1) + 1/n > ln(n+1) + (-1/n) > 0
Now 0 < 1/(n+1) < ln(1+ 1/n) < 1/n
∑ 1/(n+1) < ∑ ln(1+ 1/n) = ln(n+1)
and -ln(n) < ln(1/n) < 0 <1/n
Then ln(n+1) + 1/n = ln(n+1) + (-1/n) > 0
Now we manage to show that {a_n} has lower bound 0 and is strictly decreasing as n increases, so the sequence converges. Q.E.D.
ln(n+1) + 1/n > ln(n+1) + (-1/n) > 0
嗱，你無睇錯，烏大龜的確心大，成日犯低級錯誤
Now 0 < 1/(n+1) < ln(1+ 1/n) < 1/n
∑ 1/(n+1) < ∑ ln(1+ 1/n) = ln(n+1)
and -ln(n) < ln(1/n) < 0 <1/n
Then ln(n+1) + 1/n = ln(n+1) + (-1/n) > 0
Now we manage to show that {a_n} has lower bound 0 and is strictly decreasing as n increases, so the sequence converges. Q.E.D.
ln(n+1) + 1/n > ln(n+1) + (-1/n) > 0
嗱，你無睇錯，烏大龜的確心大，成日犯低級錯誤
ln(n+1) + 1/n > ln(n+1) + (-ln(n)) > 0
留意n係整數，所以n>0
More:
ln(n+1) + 1/n > ln(n+1) + (-1/n) > 0
嗱，你無睇錯，烏大龜的確心大，成日犯低級錯誤
ln(n+1) + 1/n > ln(n+1) + (-ln(n)) > 0
and -ln(n) = ln(1/n) < 0 <1/n
做數講求經驗，無經驗就無heuristics
烏大龜就係識concept但成日表達錯誤，
證明佢預科階段計數經驗不足，
正如計integration by part,
平唔計數就極易臨場sub錯數浪費時間，
仲未計poor presentation, 計完又唔撚proofread
這就是他高中一個A都攞唔到嘅根本原因: 粗枝大葉
and -ln(n) = ln(1/n) ≤ 0 <1/n
since min(n) = 1
畫圖就知 n > ln(n) 等同阿媽係女人，
不過在HKAL Pure Math都要循例用微積分證明